Inclusive Production of Two Rapidity-Separated Heavy Quarks As a Probe of BFKL Dynamics

Inclusive production of two rapidity-separated heavy quarks as a probe of BFKL dynamics

Andrèe Dafne Bolognino Università della Calabria & INFN - Gruppo collegato di Cosenza, I-87036 Arcavacata di Rende, Cosenza, Italy E-mail: [email protected]

Francesco Giovanni Celiberto Università di Pavia & INFN - Sezione di Pavia, I-27100 Pavia, Italy E-mail: [email protected]

Michael Fucilla Università della Calabria, I-87036 Arcavacata di Rende, Cosenza, Italy E-mail: [email protected]

Dmitry Yu. Ivanov Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia E-mail: [email protected]

Beatrice Murdaca Università della Calabria & INFN - Gruppo collegato di Cosenza, I-87036 Arcavacata di Rende, Cosenza, Italy E-mail: [email protected]

Alessandro Papa∗ Università della Calabria & INFN - Gruppo collegato di Cosenza, I-87036 Arcavacata di Rende, Cosenza, Italy E-mail: [email protected]

The inclusive photoproduction of two heavy quarks, separated by a large rapidity interval, is proposed as a new channel for the manifestation of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) dynamics. The extension to the hadroproduction case is also discussed. arXiv:1906.05940v1 [hep-ph] 13 Jun 2019

XXVII International Workshop on Deep-Inelastic Scattering and Related Subjects - DIS2019 8-12 April, 2019 Torino, Italy

∗Speaker.

c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ Inclusive production of two rapidity-separated heavy quarks Alessandro Papa

1. Introduction

Semihard processes (s Q2 Λ2 , with s the squared center-of-mass energy, Q the process QCD hard scale and ΛQCD the QCD mass scale) represent a challenge for high-energy QCD. Fixed-order perturbative calculations miss the effect of large energy logarithms, which must be resummed to all orders. The theoretical tool for this resummation is the BFKL approach [1], valid both in the n leading logarithmic approximation (LLA) (all terms (αs ln(s)) ) and in the next-to-LLA (NLA) (all n terms αs(αs ln(s)) ). In this approach, the (possibly differential) cross section factorizes into two process-dependent impact factors and a process-independent Green’s function. Only a few impact factors have been calculated with next-to-leading order accuracy: parton to parton [2], parton to forward jet [3], parton to forward hadron [4], γ∗ to light vector meson [5] and γ∗ to γ∗ [6]. They were used to build predictions for a few exclusive processes: γ∗γ∗ to two light vector mesons [7] and the γ∗γ∗ to all [8], which can be studied in future high-energy linear colliders. They enter, however, a lot of inclusive processes, accessible at LHC: Mueller-Navelet jet production [9], three and four jets, separated in rapidity [10], two identiﬁed rapidity-separated hadrons [11], forward identiﬁed light hadron and backward jet [12], forward J/Ψ-meson and backward jet [13], forward Drell-Yan pair and backward jet [14]. Here we present another possible BFKL probe: the inclusive production of two heavy quarks, separated in rapidity, in γγ collisions (photoproduction),

γ(p1) + γ(p2) Q(q1) + X + Q(q2) , (1.1) −→ where Q here stands for a c- or b-quark (see Fig. 1(left)). This process can be studied either at + e e− or in nucleus-nucleus colliders via the interaction of two quasi-real photons. Here we focus + on e e− collisions, but we brieﬂy discuss also the case of production in proton-proton collisions (hadroproduction), via a gluon-initiated subprocess.

2. Theoretical setup: photoproduction case

The impact factor relevant for the process given in (1.1) reads [15], at leading order 1

2 ααse dΦ = Q N2 1 m2R2 +~P2 z2 + z2 d2q dz, π c − q h i 1 1 ~q ~k ~q R = , ~P = + − . m2 +~q 2 − m2 + (~q ~k)2 m2 +~q 2 m2 + (~q ~k)2 − − Here α and αs denote the QED and QCD couplings, Nc the number of colors, eQ the electric charge of the heavy quark, m its mass, z and z 1 z the longitudinal fractions of the quark and antiquark ≡ − produced in the same vertex and ~k, ~q, ~k ~q the transverse momenta with respect to the photons − collision axis of Reggeized gluon, produced quark and antiquark, respectively. Similarly to the dihadron production processes (see [11]), we have

∞ dσγγ 1 = 2 C0 + 2 ∑ cos(nϕ)Cn , dy1dy2d ~q1 d ~q2 dϕ1dϕ2 (2π) | | | | " n=1 # 1In Ref. [16] the factor N2 1 was forgotten; plots and tables in the present paper take it properly into account c − and, thus, overwrite the corresponding ones in Ref. [16]. p

1 Inclusive production of two rapidity-separated heavy quarks Alessandro Papa

p1 p1 p1 Φ ~k1, ~q1,z1

   k1 k1          X   G(~k1,~k2)          k k  2 2   

Φ ~k2, ~q2,z2 p2 p2 − p2 q2

Figure 1: (Left) Heavy-quark pair photoproduction. (Right) BFKL factorization: crosses denote the tagged quarks, whose momenta are not integrated over for getting the cross section.

2 2 2 Table 1: C0 [pb] vs. ∆Y for qmin = 0 GeV and √s = 200 GeV; C stands for µR/(s1s2), with s1,2 = m1,2 +~q1,2.

LLA LLA LLA NLA NLA NLA ∆Y Box qq¯ C = 1/2 C = 1 C = 2 C = 1/2 C = 1 C = 2 1.5 98.26 415.0(1.3) 65.24(31) 28.94(14) 16.96(10) 11.237(73) 10.289(74) 2.5 42.73 723.7(2.1) 88.64(36) 34.58(17) 17.580(91) 9.581(57) 8.504(56) 3.5 14.077 1203.4(3.4) 113.33(43) 39.01(16) 18.522(92) 7.989(43) 6.637(36) 4.5 3.9497 1851.6(5.0) 133.64(52) 40.42(19) 18.412(90) 6.210(31) 4.893(25) 5.5 0.9862 2559.4(7.1) 140.23(55) 37.18(17) 16.971(83) 4.329(21) 3.138(15)

1 where ϕ = ϕ1 ϕ2 π, with ϕ1 2, ~q1 2 and y1 2, respectively, the azimuthal angles, the transverse − − , , , momenta and the rapidities of the produced quarks. The Cn coefﬁcients encode the two leading- order impact factors and the NLA BFKL Green’s function (see Fig. 1(right)). For brevity, the expression for Cn is not presented here; it can be found in Ref. [16], to which we refer for all details concerning the present paper. + For the process initiated by e e− collisions, we must take into account the ﬂux of quasi-real photons dn/dx emitted by each of the two colliding particles. The cross section, differential in the rapidity gap ∆Y between the two tagged heavy quarks, reads then

q q y(1) y(2) dσ + max max max max e e− = dq dq dy dy δ (y y ∆Y) 1 2 (1) 1 (2) 2 1 2 d (∆Y) qmin qmin ymax ymax − − Z Z Z− Z− 1 1 dn1 dn2 (1) (2) ymax y dx1 ymax+y dx2 dσγγ , (2.1) × e− − 1 dx e− 2 dx Z 1 Z 2 (1) s (2) s with ymax = ln 2 2 and ymax = ln 2 2 , where s is the squared center-of-mass energy of the m1+~q1 m2+~q2 + colliding e e−qpair. In the followingq we will present results for the integrated azimuthal coefﬁ- cients Cn, deﬁned through

∞ d + 1 σe e− = 2 C0 + 2 ∑ cos(nϕ)Cn . d (∆Y)dϕ1dϕ2 (2π) " n=1 #

2 Inclusive production of two rapidity-separated heavy quarks Alessandro Papa

2 2 2 Table 2: C0 [pb] vs. ∆Y for qmin = 0 GeV and √s = 3 TeV; C stands for µR/(s1s2), with s1,2 = m1,2 +~q1,2.

LLA LLA LLA NLA NLA NLA ∆Y Box qq¯ C = 1/2 C = 1 C = 2 C = 1/2 C = 1 C = 2 1.5 280.98 10.893(49) 103 530.8(2.4) 195.54(88) 99.57(89) 58.34(58) 52.17(58) · 3.5 48.93 54.84(14) 103 1568.9(7.6) 439.4(2.1) 184.5(1.1) 65.22(50) 54.38(47) · 5.5 4.9819 254.88(57) 103 4409(19) 930.6(4.2) 380.2(1.8) 75.83(53) 55.22(36) · 7.5 0.4318 1041.7(2.1) 103 10.921(44) 103 1743.1(8.3) 756.3(3.5) 81.94(44) 51.48(27) · · 9.5 0.0323 3429.0(7.8) 103 21.530(80) 103 2618(12) 1267.0(6.1) 72.73(36) 38.86(19) · · 10.5 0.0081 5468(14) 103 26.23(10) 103 2761(12) 1443.3(7.2) 59.97(30) 29.21(14) · ·

3. Numerical analysis

2 We consider only the case of c-quark with mass m = 1.2 GeV/c and ﬁx qmax = 10 GeV and qmin = 0, 1, 3 GeV. We take √s = 200 GeV, as in LEP2, with 1 < ∆Y < 6, and √s = 3 TeV, as in + the future e e− CLIC linear accelerator, with 1 < ∆Y < 11. In Tables 1-2 we show pure LLA and NLA BFKL predictions for C0 with qmin = 0 GeV and √s = 200 GeV and 3 TeV, respectively, and compare them with the exclusive photoproduction of a cc¯ pair, given by two “box” diagrams. We see that at LEP2 energies the “box” cross section dominates, but at CLIC energies BFKL takes over.

Results for C0, R10, and R20 C2/C0 with qmin = 1, 3 GeV and √s = 200 GeV and 3 TeV are ≡ shown in Fig. 2. We see that the cross section increases from LEP2 to CLIC energies and decreases from LLA to NLA. Azimuthal correlations are in all cases much smaller than one and decrease when ∆Y increases, as it must be due to the larger emission of undetected partons. Moreover, the inclusion of NLA effects increases the correlations, which can only be explained with the larger suppression of C0 with respect to C1,2 when these effects are included.

4. Theoretical setup: hadroproduction case

The hadroproduction case can be studied in a similar fashion as the photoproduction one, with the role of photons played by gluons and the photon ﬂux replaced by the gluon parton distribution function in the proton. The differential impact factor in this case takes the form

d 2 N2 1 2 N2 Φ αs c 2 ¯ 2 ~ ~¯ 2 2 c 2 ¯ ~~¯ 2 2 2 = − m (R + R) + P + P z + z¯ 2 2 m RR + PP z + z¯ , d q dz 2πNc − N 1 p c − 1 1 ~q ~kz ~q R = , ~P = + − , m2 +~q2 − m2 + (~q ~kz)2 m2 +~q2 m2 + (~q ~kz)2 − − 1 1 ~k ~q ~kz ~q R¯ = − + , ~P¯ = − − . m2 + (~q ~k)2 m2 + (~q ~kz)2 m2 + (~k ~q)2 − m2 + (~q ~kz)2 − − − − Color and coupling prefactors enhance hadroproduction cross section by some 103 with respect to photoproduction, but photon ﬂux dn/dx dominates over g(x) for x 0 and x 1 so that it is not → → easy to estimate the size of the cross section without a detailed numerical analysis.

3 Inclusive production of two rapidity-separated heavy quarks Alessandro Papa

(cc, cc)

-1 10 MS scheme 2 -2 s = (200 GeV) 10

q1, q2 > 3 GeV 2 1/2 -2 µ = C (s s ) 10 R 1 2 -3 10 [nb] [nb] 0 0 C C -3 10 (cc, cc) -4 MS scheme 10 LLA - C = 1/2 2 LLA - C = 1/2 LLA - C = 1 s = (200 GeV) LLA - C = 1 -4 LLA - C = 2 LLA - C = 2 10 NLA - C = 1/2 q1, q2 > 1 GeV NLA - C = 1/2 NLA - C = 1 NLA - C = 1 2 1/2 -5 NLA - C = 2 µR = C (s1s2) 10 NLA - C = 2

1 2 3 4 5 6 1 2 3 4 5 6 ∆Y ∆Y LLA - q = 1 GeV LLA - q = 1 GeV min (cc, cc) min (cc, cc) LLA - qmin = 3 GeV LLA - qmin = 3 GeV NLA - q = 1 GeV NLA - q = 1 GeV 0.30 min MS scheme 0.30 min MS scheme NLA - q = 3 GeV 2 NLA - q = 3 GeV 2 min s = (200 GeV) min s = (200 GeV)

0.25 q1, q2 > qmin 0.25 q1, q2 > qmin 2 1/2 2 1/2 µR = (s1s2) µR = (s1s2)

0 0.20 0 0.20 /C /C 1 2 = C = C

10 0.15 20 0.15 R R

0.10 0.10

0.05 0.05

0.00 0.00 1 2 3 4 5 6 1 2 3 4 5 6 ∆Y ∆Y

LLA - qmin = 1 GeV LLA - q = 3 GeV (cc, cc) 0 min 10 NLA - q = 1 GeV 0.30 min MS scheme NLA - q = 3 GeV 2 min s = (3 TeV) -1 10 0.25 q1, q2 > qmin 2 1/2 µ = (s s ) -2 R 1 2

10 0 0.20 /C 1 [nb]

0 -3 10 = C C 10 0.15 R (cc, cc) -4 10 MS scheme 0.10 LLA - qmin = 1 GeV 2 LLA - q = 3 GeV s = (3 TeV) -5 min 10 NLA - q = 1 GeV min q1, q2 > qmin 0.05 NLA - q = 3 GeV 2 1/2 min µ = (s s ) -6 R 1 2 10 0.00 2 4 6 8 10 2 4 6 8 10 ∆Y ∆Y

Figure 2: ∆Y-dependence of C0, R10, and R20 for qmin = 1, 3 GeV, √s = 200 GeV and 3 TeV, and for different 2 2 2 values of C = µR/√s1s2, with s1,2 = m1,2 +~q1,2.

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